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Unformatted text preview: Derivation of the Lorentz Transformation Lecture note for course Phys171H “Introductory Physics: Mechanics and Relativity” written by Victor Yakovenko Department of Physics, University of Maryland, College Park 15 November 2004 In most textbooks, the Lorentz transformation is derived from the two postulates: the equivalence of all inertial reference frames and the invariance of the speed of light. However, the most general transformation of space and time coordinates can be derived using only the equivalence of all inertial reference frames and the symmetries of space and time. The general transformation depends on one free parameter with the dimensionality of speed, which can be then identified with the speed of light c . This derivation uses the group property of the Lorentz transformations, which means that a combination of two Lorentz transformations also belongs to the class Lorentz transformations. The derivation can be compactly written in matrix form. However, for those not familiar with matrix notation, I also write it without matrices. 1) Let us consider two inertial reference frames O and O . The reference frame O moves relative to O with velocity v in along the x axis. We know that the coordinates y and z perpendicular to the velocity are the same in both reference frames: y = y and z = z . So, it is sufficient to consider only transformation of the coordinates x and t from the reference frame O to x = f x ( x,t ) and t = f t ( x,t ) in the reference frame O . From translational symmetry of space and time, we conclude that the functions f x ( x,t ) and f t ( x,t ) must be linear functions. Indeed, the relative distances between two events in one reference frame must depend only on the relative distances in another frame: x 1- x 2 = f x ( x 1- x 2 ,t 1- t 2 ) , t 1- t 2 = f t ( x 1- x 2 ,t 1- t 2 ) . (1) Because Eq. (1) must be valid for any two events, the functions f x ( x,t ) and f t ( x,t ) must be linear functions. Thus x = Ax + Bt, (2) t = Cx + Dt, (3) where A ,...
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This note was uploaded on 10/05/2009 for the course PHYSICS 7C taught by Professor Lin during the Spring '08 term at Berkeley.
- Spring '08
- Quantum Physics